From Rotman's Algebraic Topology, I have a question about what the proof is saying concerning the below theorem:
If $K$ is a connected simplicial complex with basepoint $p$, then $\pi(K,p) \approx G_{K,T}.$
$\pi(K,p)$ is the group of equivalence classes of paths in $K$ with basepoint $p$, and $G_{K,T}$ is defined below as:
And since this is a presentation, it must be the case that $G_{K,T} \approx F/N$, where $F$ is the free group with basis of the generators in the above definition and $N$ defined as the normal subgroup generated by the relations in the above definition.
In the picture, $\alpha_v$ is defined as a path from $p \mapsto v$.
In the attached image in the first paragraph, why does he go from talking about a map between $F$ and $\pi(K,p)$ and then state the same map defines a homomorphism between $F/R=G_{K,T}$? Is he talking about two different maps or the same map? And why is he saying that $F/R=G_{K,T}$ since they're only related via isomorphism? I would think it would be the first isomorphism theorem, but $\phi(F)$ doesn't seem to be onto.